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<h1>Using Map Projections in GrADS</h1>

It is important to understand the distinction between the two
uses of map projections when creating GrADS displays of your
data:<p>

<ul>
<li>projection of the data (preprojected grids);<br>
<li>projection of the display.</ul><p>

GrADS supports two types of data grids:<p>
<ul>
<li><code>lon/lat</code> grids (and not necessarily regular,
e.g., gaussian);<br>
<li>preprojected grids.</ul><p>
<ul>
<a href="#pre">Using Preprojected Grids</a><br>
<a href="#proj">GrADS Display Projections</a><br>
<a href="#summary">Summary and Plans</a></ul><br>
<hr>
<p>

<a name="pre"><h2><u>Using Preprojected Grids</u></h2></a>
<ul>
<a href="#polar">Polar Stereo Preprojected Data</a><br>
<a href="#lambert">Lambert Conformal Preprojected Data</a><br>
<a href="#eta">NMC Eta model</a><br>
<a href="#nmc">NMC high accuracy polar stereo for SSM/I 
data</a><br>
<a href="#csu">CSU RAMS Oblique Polar Stereo Grids</a><br>
<a href="#pit">Pitfalls when using preprojected
data</a></ul><br>
<br>
<ul><b>Preprojected</b> data are data <b>already</b> on a map
projection. GrADS
supports four types of preprojected data:<p>

<ol>
<li>N polar stereo (NMC model projection); 
<li>S polar stereo (NMC model projection) ; 
<li>Lambert Conformal (originally for Navy NORAPS model); 
<li>NMC eta model (unstaggered). 
<li>More precise N and S polar stereo (hi res SSM/I data) 
<li>Colorado State University RAMS model (oblique polar stereo;
beta)</ol><p>

When preprojected grids are opened in GrADS, bilinear
interpolation constants are calculated and all date are displayed
on an internal GrADS lat/lon grid defined by the
<code>xdef</code> and <code>ydef</code>
card in the data description or <code>.ctl</code> file (that's
why it takes
longer to "open" a preprojected grid data set).<p>

It is very important to point out that the internal GrADS grid
can be any grid as it is completely independent of the
preprojected data grid.  Thus, there is nothing
stopping you
displaying preprojected data on a very high res
lon/lat grid
(again, defined in the <code>.ctl</code> by <code>xdef</code> 
and <code>ydef</code>). In fact, you
could create and open multiple .ctl files with different
resolutions and/or regions which pointed to the same
preprojected
data file.<p>

When you do a <a
href="gradcomddisplay.html"><code>display</code></a>
(i.e., get a grid of data), the
preprojected data are bilinearly interpolated to
the GrADS
internal lat/lon grid.  For
preprojected
scalar fields (e.g., 500
mb heights), the display is adequate and the precision of the
interpolation can be controlled by <code>xdef</code> and
<code>ydef</code> to define a
higher spatial resolution grid.<p>

The big virtue of this approach is that all built in GrADS
analytic functions (e.g., <a
href="gradfuncaave.html"><code>aave</code></a>,
<a href="gradfunchcurl.html"><code>hcurl</code></a>...)
continue to work even
though the data were not originally on a lon/lat grid.  The
downside is that you are not looking directly at your data on a
geographic map.  However, one could always define a .ctl file
which simply opened the data file as i,j data and displayed
without the map (<a href="gradcomdsetmpdraw.html"><code>set
mpdraw</a> off</code>). So, in my opinion, this
compromise is not that limiting even if as a modeller you wanted
to look at the grid--you just don't get the map background.<p>

<code>Preprojected vector fields</code> are a little trickier,
depending on
whether the vector is defined relative to the data grid or
relative to the Earth.  For example, NMC polar stereo grids use
winds relative to the data grid and thus must be rotated to the
internal GrADS lat/lon grid (again defined in the
<code>.ctl</code> file by
the <code>xdef</code> and <code>ydef</code> cards).<p>

The only potential problem with working with preprojected data
(e.g., Lambert Conformal model data) is defining the projection
for GrADS. This is accomplished using a <code>pdef</code> card
in the data
descriptor <code>.ctl</code> file.
</ul><br><br>

<a name="polar"><b><i>Polar Stereo Preprojected Data (coarse
accuracy for NMC
Models)</i></b><p>
<ul>
Preprojected data on a polar stereo projection (N and S) is
defined as at NMC.  For the NCEP model GRIB data distributed
via anon ftp from ftp.ncep.noaa.gov, the <code>pdef</code> card
is:<p>
<ul>
<code>
pdef isize jsize projtype ipole jpole lonref gridinc<br>
pdef 53 45 nps 27 49 -105 190.5
</code>
</ul><p>

where,<p>

<ul><code>ipole</code> and <code>jpole</code> are the (i,j) of
the pole referenced from the lower left corner at (1,1) and <code>gridinc</code> is the dx
in km.</ul><p>

The relevant GrADS source is:<p>

<code>
void w3fb04 (float alat, float along, float xmeshl, float
orient,
float *xi, float *xj) {<br>
 /* <br>
C <br>
C SUBPROGRAM: W3FB04     LATITUDE, LONGITUDE TO GRID
COORDINATES <br>
C   AUTHOR: MCDONELL,J.   
  ORG: W345       DATE: 90-06-04 <br>
C <br>
C ABSTRACT: CONVERTS THE
COORDINATES OF A LOCATION ON EARTH FROM THE <br>
C   NATURAL
COORDINATE SYSTEM OF LATITUDE/LONGITUDE TO THE GRID (I,J) <br>
C  
COORDINATE SYSTEM OVERLAID ON A POLAR STEREOGRAPHIC MAP PRO
<br>
C  
JECTION TRUE AT 60 DEGREES N OR S LATITUDE. W3FB04 IS THE
REVERSE<br>
C   OF W3FB05. <br>
C <br>
C PROGRAM HISTORY LOG: <br>
C   77-05-01  J.
MCDONELL<br>
C   89-01-10  R.E.JONES   CONVERT TO MICROSOFT FORTRAN 4.1
<br>
C  
90-06-04  R.E.JONES   CONVERT TO SUN FORTRAN 1.3 <br>
C
93-01-26 B.
Doty     converted to <br>
C <br>C <br>C USAGE:  CALL W3FB04 (ALAT,
ALONG,
XMESHL, ORIENT, XI, XJ) <br>C <br>C   INPUT VARIABLES: <br>C
NAMES 
INTERFACE DESCRIPTION OF VARIABLES AND TYPES <br>C     ------
--------- ----------------------------------------------- <br>C    
ALAT   ARG LIST  LATITUDE IN DEGREES (<0 IF SH) <br>C     ALONG
ARG
LIST  WEST LONGITUDE IN DEGREES <br>C     XMESHL ARG LIST  MESH
LENGTH OF GRID IN KM AT 60 DEG LAT(<0 IF SH) <br>C                  
(190.5 LFM GRID, 381.0 NH PE GRID,-381.0 SH PE GRID) <br>C
ORIENT
ARG LIST  ORIENTATION WEST LONGITUDE OF THE GRID <br>C                
  (105.0 LFM GRID, 80.0 NH PE GRID, 260.0 SH PE GRID) <br>C
<br>C  
OUTPUT VARIABLES: <br>C     NAMES  INTERFACE DESCRIPTION OF
VARIABLES
AND TYPES <br>C     ------ ---------
----------------------------------------------- <br>C     XI
ARG
LIST  I OF THE POINT RELATIVE TO NORTH OR SOUTH POLE <br>C
XJ    
ARG LIST  J OF THE POINT RELATIVE TO NORTH OR SOUTH POLE <br>C
<br>C  
SUBPROGRAMS CALLED: <br>C     NAMES                                   
               LIBRARY <br>C    
------------------------------------------------------- --------
<br>C     COS SIN                                                
SYSLIB <br>C <br>C   REMARKS: ALL PARAMETERS IN THE CALLING
STATEMENT
MUST BE <br>C     REAL. THE RANGE OF ALLOWABLE LATITUDES IS
FROM A
POLE TO <br>C     30 DEGREES INTO THE OPPOSITE HEMISPHERE.
<br>C THE
GRID USED IN THIS SUBROUTINE HAS ITS ORIGIN (I=0,J=0) <br>C
AT
THE POLE IN EITHER HEMISPHERE, SO IF THE USER'S GRID HAS ITS
<br>C    
ORIGIN AT A POINT OTHER THAN THE POLE, A TRANSLATION IS NEEDED
<br>C  
  TO GET I AND J. THE GRIDLINES OF I=CONSTANT ARE PARALLEL TO A
<br>C 
   LONGITUDE DESIGNATED BY THE USER. THE EARTH'S RADIUS IS TAKEN
C     TO BE 6371.2 KM. <br>C <br>C ATTRIBUTES: <br>C
LANGUAGE: SUN FORTRAN
1.4 C   MACHINE:  SUN SPARCSTATION 1+ <br>C*/ 
<br>static float
radpd =
0.01745329; <br>
static float earthr = 6371.2;<p>

float re,xlat,wlong,r; <br>
&nbsp;&nbsp;&nbsp;re    = (earthr * 1.86603) / xmeshl;<br>
&nbsp;&nbsp;&nbsp;xlat
=  alat * radpd; <br>
&nbsp;&nbsp;&nbsp;if (xmeshl>0.0) { <br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;wlong =
(along + 180.0 -
orient) * radpd; <br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;r     =
(re * cos(xlat)) / (1.0 +
sin(xlat));
<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;*xi    =
r * sin(wlong); <br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;*xj = r *
cos(wlong);<p>

&nbsp;&nbsp;&nbsp;} else { <br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;re    = -re; <br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;xlat =
-xlat;
<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;wlong = (along - orient) *
radpd; <br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;r     =
(re * cos(xlat)) / (1.0+ sin(xlat)); <br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;*xi =
r *
sin(wlong); <br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;*xj   =
-r * cos(wlong); <br>
&nbsp;&nbsp;&nbsp;} <br>
}
</code></ul>
<br>
<br>

<a name="lambert"><b><i>Lambert Conformal Preprojected
Data</i></b></a><p>
<ul>

The Lambert Conformal projection (lcc) was implemented for the
U.S. Navy's limited area model NORAPS.  Thus, to work with your
lcc data you must express your grid in the context of the Navy
lcc grid.  NMC has been able to do this for their AIWIPS grids
and the Navy definition should be general enough for others.<p>

A typical NORAPS Lambert-Conformal grid is described below,
including the C code which sets up the internal
interpolation.<p>

The <code>.ctl</code> file is:<p>

<ul>
<code>
dset ^temp.grd <br>
title NORAPS DATA TEST <br>
undef 1e20 <br>
pdef 103 69 lcc 30 -88 51.5 34.5 20 40 -88 90000 90000 <br>
xdef 180 linear -180 1.0<br>
ydef 100 linear  -10 1.0 <br>
zdef  16 levels 1000 925 850 700 500 400 300 250 200 150 100 70
50 30 20 10 <br>
tdef   1 linear 00z1jan94 12hr<br>
vars 1 <br>
&nbsp;&nbsp;&nbsp;t 16 0 temp <br>
endvars</code></ul><p>

where,<p>

<ul>
<code>103&nbsp;&nbsp;&nbsp;</code>= #pts in x <br>
<code>69&nbsp;&nbsp;&nbsp;&nbsp;</code>= #pts in y <br>
<code>lcc&nbsp;&nbsp;&nbsp;</code>= Lambert-Conformal <br>
<code>30&nbsp;&nbsp;&nbsp;&nbsp;</code>= lat of ref point <br>
<code>88&nbsp;&nbsp;&nbsp;&nbsp;</code>= lon of ref point (E is positive, W is negative) <br>
<code>51.5&nbsp;&nbsp;</code>= i of ref point <br>
<code>34.5&nbsp;&nbsp;</code>= j of ref point <br>
<code>20&nbsp;&nbsp;&nbsp;&nbsp;</code>= S true lat <br>
<code>40&nbsp;&nbsp;&nbsp;&nbsp;</code>= N true lat <br>
<code>88&nbsp;&nbsp;&nbsp;&nbsp;</code>= standard lon <br>
<code>90000&nbsp</code>= dx in M <br>
<code>90000&nbsp</code>= dy in M
</ul><p>

Otherwise, it is the same as other GrADS files.<p>

<b>Note</b> - the <code>xdef/ydef</code> apply to the
<code>lon/lat</code> grid GrADS internally interpolates to and
can be anything...<p>

The GrADS source which maps <code>lon/lat</code> of the GrADS
internal <code>lon/lat</code>
grid to <code>i,j</code> of the preprojected grid is:<p>

<code>
<pre>
/* Lambert Conformal conversion */
void ll2lc (float *vals, float grdlat, float grdlon, 
float *grdi, float *grdj) {
/*  Subroutine to convert from lat-lon to Lambert Conformal i,j.
Provided by NRL Monterey; converted to C 6/15/94.
c                SUBROUTINE: ll2lc
c
c                PURPOSE: To compute i- and j-coordinates of a specified
c                         grid given the latitude and longitude points. 
c                         All latitudes in this routine start 
c                         with -90.0 at the south pole and increase 
c                         northward to +90.0 at the north pole.  The 
c                         longitudes start with 0.0 at the Greenwich 
c                         meridian and increase to the east, so that 
c                         90.0 refers to 90.0E, 180.0 is the inter- 
c                         national dateline and 270.0 is 90.0W. 
c
c                INPUT VARIABLES: 
c 
c   vals+0         reflat: latitude at reference point (iref,jref) 
c
c   vals+1         reflon: longitude at reference point (iref,jref) 
c 
c   vals+2         iref:   i-coordinate value of reference point 
c 
c   vals+3         jref:   j-coordinate value of reference point 
c 
c   vals+4         stdlt1: standard latitude of grid 
c 
c   vals+5         stdlt2: second standard latitude of grid (only required 
c                  if igrid = 2, lambert conformal) 
c 
c   vals+6         stdlon: standard longitude of grid (longitude that 
c                          points to the north) 
c 
c   vals+7         delx:   grid spacing of grid in x-direction 
c                          for igrid = 1,2,3 or 4, delx must be in meters 
c                          for igrid = 5, delx must be in degrees 
c 
c   vals+8         dely:   grid spacing (in meters) of grid in y-direction 
c                          for igrid = 1,2,3 or 4, delx must be in meters 
c                          for igrid = 5, dely must be in degrees 
c 
c                  grdlat: latitude of point (grdi,grdj) 
c 
c                  grdlon: longitude of point (grdi,grdj)
c 
c                  grdi:   i-co ordinate(s) that this routine will generate 
c                          information for 
c 
c                  grdj:   j-coordinate(s) that this routine will generate 
c                          information for 
c 
*/

   float pi, pi2, pi4, d2r, r2d, radius, omega4; 
   float gcon,ogcon,ahem,deg,cn1,cn2,cn3,cn4,rih,xih,yih,rrih,check; 
   float alnfix,alon,x,y; 
   pi = 4.0*atan(1.0); 
   pi2 = pi/2.0; 
   pi4 = pi/4.0;
   d2r = pi/180.0; 
   r2d = 180.0/pi; 
   radius = 6371229.0; 
   omega4 = 4.0*pi/86400.0;

/*mf -------------- mf*/ 
/*case where standard lats are the same */ 
  if(*(vals+4) == *(vals+5)) { 
    gcon = sin(*(vals+4)*d2r); 
  } else { 
    gcon = (log(sin((90.0-*(vals+4))*d2r))
    log(sin((90.0-*(vals+5))*d2r)))
    /(log(tan((90.0-*(vals+4))*0.5*d2r))
    log(tan((90.0-*(vals+5))*0.5*d2r))); 
  } 
/*mf -------------- mf*/
  ogcon = 1.0/gcon; 
  ahem = fabs(*(vals+4))/(*(vals+4)); 
  deg = (90.0-fabs(*(vals+4)))*d2r; 
  cn1 = sin(deg); 
  cn2 = radius*cn1*ogcon;
  deg = deg*0.5; 
  cn3 = tan(deg); 
  deg = (90.0-fabs(*vals))*0.5*d2r;
  cn4 = tan(deg); 
  rih = cn2*pow((cn4/cn3),gcon); 
  deg = (*(vals+1)-*(vals+6))*d2r*gcon; 
  xih = rih*sin(deg); 
  yih = -rih*cos(deg)*ahem; 
  deg = (90.0-grdlat*ahem)*0.5*d2r; 
  cn4 = tan(deg); 
  rrih = cn2*pow((cn4/cn3),gcon); 
  check = 180.0-*(vals+6); 
  alnfix = *(vals+6)+check; 
  alon = grdlon+check;
  while (alon<0.0) alon = alon+360.0; 
  while (alon>360.0) alon = alon-360.0; 
  deg = (alon-alnfix)*gcon*d2r; 
  x = rrih*sin(deg); 
  y = -rrih*cos(deg)*ahem; 
  *grdi = *(vals+2)+(x-xih)/(*(vals+7)); 
  *grdj = *(vals+3)+(y-yih)/(*(vals+8)); 
}
</pre></code></ul>
<br>
<br><a name="eta"><b><i>NMC Eta model (unstaggered
grids)</i></b></a><p>
<ul>

The NMC eta model "native" grid is awkward to work with because
the variables are on staggered (e.g., the grid for winds is not
the same as the grid for mass points) <i>and</i> non rectangular (number
of points in i is <i>not</i> constant with j) grids.  Because any
contouring of irregularly gridded data involves interpolation at
some point, NMC creates "unstaggered"  eta model fields for
practical application programs such as GrADS.  In the unstaggered
grids all variables are placed on a common <i>and</i> rectangular grid
(the mass points).<p>

Wind rotation has also been added so that vector data will be
properly displayed.<p>

The pdef card for a typical eta model grid is:<p>

<ul>
<code>pdef 181 136 eta.u -97.0 41.0 0.38888888 0.37037037</code><p>

<code>181</code>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; = #pts in x <br>
<code>136</code>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;    = #pts in y <br>
<code>eta.u</code>&nbsp;&nbsp;&nbsp;&nbsp;  = eta grid, unstaggered<br>
<code>-97.0</code>&nbsp;&nbsp;&nbsp;&nbsp;  = lon of ref point (E is positive in GrADS, W
is negative)
[deg] <br>
<code>41.0</code>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; = lat of ref point [deg] <br>
<code>0.3888</code>&nbsp;&nbsp; = dlon [deg] <br>
<code>0.37037</code> = dlat [deg]</ul><p>

The source code in GrADS for the lon,lat -> i,j mapping is:<p>
<pre>
void ll2eg (int im, int jm, float *vals,  float grdlon, float grdlat,
      float *grdi, float *grdj, float *alpha) {

/*  Subroutine to convert from lat-lon to NMC eta i,j.

    Provided by Eric Rogers NMC; converted to C 3/29/95 by Mike Fiorino.

c                SUBROUTINE: ll2eg 
c 
c                PURPOSE: To compute i- and j-coordinates of a specified 
c                         grid given the latitude and longitude points.
c                         All latitudes in this routine start 
c                         with -90.0 at the south pole and increase 
c                         northward to +90.0 at the north pole.  The 
c                         longitudes start with 0.0 at the Greenwich 
c                         meridian and increase to the east, so that 
c                         90.0 refers to 90.0E, 180.0 is the inter- 
c                         national dateline and 270.0 is 90.0W. 
c 
c                INPUT VARIABLES: 
c 
c   vals+0         tlm0d: longitude of the reference center point 
c 
c   vals+1         tph0d: latitude of the reference center point 
c 
c   vals+2         dlam:  dlon grid increment in deg 
c 
c   vals+3         dphi:  dlat grid increment in deg 
c 
c 
c                  grdlat: latitude of point (grdi,grdj) 
c 
c                  grdlon: longitude of point (grdi,grdj) 
c 
c                  grdi:   i-coordinate(s) that this routine will generate 
c                          information for 
c 
c                  grdj:   j-coordinate(s) that this routine will generate 
c                          information for 
c

*/

   float pi,d2r,r2d, earthr;   float tlm0d,tph0d,dlam,dphi;  
   float 
    phi,lam,lame,lam0,phi0,lam0e,cosphi,sinphi,sinphi0,cosphi0,sinlam r,cos 
    lamr;
   float x1,x,y,z,bigphi,biglam,cc,num,den,tlm,tph;

   int idim,jdim;

   pi=3.141592654;

   d2r=pi/180.0;   
   r2d=1.0/d2r;   
   earthr=6371.2;

   tlm0d=-*(vals+0); /* convert + W to + E, the grads standard for
    longitude */   
   tph0d=*(vals+1);   
   dlam=(*(vals+2))*0.5;  
   dphi=(*(vals+3))*0.5;

   /* grid point and center of eta grid trig */

   /* convert to radians */

   phi    = grdlat*d2r;   
   lam    = -grdlon*d2r; /* convert + W to + E, the grads standard for 
    longitude */   
   lame   = (grdlon)*d2r;

   phi0   = tph0d*d2r;   
   lam0   = tlm0d*d2r;   
   lam0e  = ( 360.0 + *(vals+0) )*d2r;

   /* cos and sin */

   cosphi = cos(phi);   
   sinphi = sin(phi);

   sinphi0 = sin(phi0);   
   cosphi0 = cos(phi0);

   sinlamr=sin(lame-lam0e);   
   coslamr=cos(lame-lam0e);

   x1     = cosphi*cos(lam-lam0);   
   x      = cosphi0*x1+sinphi0*sinphi;   
   y      = -cosphi*sin(lam-lam0);   
   z      = -sinphi0*x1+cosphi0*sinphi;

   /* params for wind rotation alpha */

   cc=cosphi*coslamr;   
   num=cosphi*sinlamr;  
   den=cosphi0*cc+sinphi0*sinphi;

   tlm=atan2(num,den);

   /* parms for lat/lon -> i,j */

   bigphi = atan(z/(sqrt(x*x+y*y)))*r2d;   
   biglam = atan(y/x)*r2d;

   idim = im*2-1;   
   jdim = jm*2-1 ;

   *grdi  = (biglam/dlam)+(idim+1)*0.5;   
   *grdj  = (bigphi/dphi)+(jdim+1)*0.5;   
   *grdi  = (*grdi+1)*0.5-1;   
   *grdj  = (*grdj+1)*0.5-1;

   *alpha = asin( ( sinphi0*sin(tlm)) / cosphi ) ;
 
/*   printf("qqq %6.2f %6.2f %6.2f %6.2f %g %g %g %g\n",    
       grdlon,grdlat,*grdi,*grdj,*alpha,tlm*r2d,cosphi,sinphi0); 
*/

}
</code></pre></ul>
<br>
<br>

<a name="nmc"><b><i>NMC high accuracy polar stereo for SSM/I
data</i></b></a><p>
<ul>
The polar stereo projection used by the original NMC models is
not very precise because it assumes the earth is round
(eccentricity = 0).  While this approximation was reasonable for
coarse resolution NWP models, it is inadequate to work with
higher resolution data such as SSM/I.<p>

<i>Wind rotation has not been implemented!!!  Use only for scalar
fields.</i><p>

<ul><code>pdef ni nj pse slat slon polei polej dx dy sgn</code><p>

<code>ni</code>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  = # points in x
<br>
<code>nj</code>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; = # points in y <br>
<code>slat</code>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; = absolute value of the standard latitude
<br>
<code>slon</code>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; = absolute value of the standard longitude
<br>
<code>pse</code>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  = polar stereo,
"eccentric"<br>
<code>polei</code>&nbsp;&nbsp;&nbsp;  = x index position of the pole (where (0,0) is the
index of the first point vice the more typical (1,1) ) <br>
<code>polej</code>&nbsp;&nbsp;&nbsp;  = y index position of the pole (where (0,0) is the
index of the first point vice the more typical (1,1) ) <br>
<code>dx</code>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; = delta x in km <br>
<code>dy</code>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; = delta y in km <br>
<code>sgn</code>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; = 1 for N polar stereo and -1
for S polar
stereo</ul><p>

Source code in GrADS for the lon,lat -> i,j mapping:<p>
<pre>
</code>
void ll2pse (int im, int jm, float *vals, float lon, float lat,   
       float *grdi, float *grdj) {

   /* Convert from geodetic latitude and longitude to polar stereographic      
      grid coordinates.  Follows mapll by V. J. Troisi.         */   
   /* Conventions include that slat and lat must be absolute values */   
   /* The hemispheres are controlled by the sgn parameter */   
   /* Bob Grumbine 15 April 1994. */

   const rearth = 6738.273e3;   
   const eccen2 = 0.006693883;  
   const float pi = 3.141592654;

   float cdr, alat, along, e, e2;
   float t, x, y, rho, sl, tc, mc;
   float slat,slon,xorig,yorig,sgn,polei,polej,dx,dy;

   slat=*(vals+0);
   slon=*(vals+1);
   polei=*(vals+2);  
   polej=*(vals+3);
   dx=*(vals+4)*1000;
   dy=*(vals+5)*1000;  
   sgn=*(vals+6);

   xorig = -polei*dx;
   yorig = -polej*dy;

   /*printf("ppp %g %g %g %g %g %g
   %g\n",slat,slon,polei,polej,dx,dy,sgn);*/

   cdr   = 180./pi;
   alat  = lat/cdr;
   along = lon/cdr;
   e2    = eccen2;
   e     = sqrt(eccen2);

   if ( fabs(lat) > 90.)  {
     *grdi = -1;
     *grdj = -1;    
     return;
   }
   else {
     t = tan(pi/4. - alat/2.) /
       pow( (1.-e*sin(alat))/(1.+e*sin(alat)) , e/2.);

     if ( fabs(90. - slat) < 1.E-3) {
       rho = 2.*rearth*t/
     pow( pow(1.+e,1.+e) * pow(1.-e,1.-e) , e/2.);
       }
       else {
         sl = slat/cdr;
         tc = tan(pi/4.-sl/2.) /
         pow( (1.-e*sin(sl))/(1.+e*sin(sl)), (e/2.) );
         mc = cos(sl)/ sqrt(1.-e2*sin(sl)*sin(sl) );
         rho = rearth * mc*t/tc;
       }

       x = rho*sgn*cos(sgn*(along+slon/cdr));
       y = rho*sgn*sin(sgn*(along+slon/cdr));

       *grdi = (x - xorig)/dx+1;
       *grdj = (y - yorig)/dy+1;

       /*printf("ppp (%g %g) (%g %g %g) %g
     %g\n",lat,lon,x,y,rho,*grdi,*grdj);*/

       return;
   }

}
</code></pre></ul>
<br>
<br>

<a name="csu"><b><i>CSU RAMS Oblique Polar Stereo
Grids</i></b></a><p>
<ul>

The CSU RAMS model uses an oblique polar stereo projection.  This
projection is still being tested...<p>

<ul>
<code>
pdef 26 16 ops 40.0 -100.0 90000.0 90000.0 14.0 9.0 180000.0
180000.0</code><p>

<code>26</code>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
= #pts in x <br>
<code>16</code>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
= #pts in y <br>
<code>ops</code>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;   =
oblique polar
stereo<br>
<code>40.0</code>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  = lat of ref
point (14.0, 9.0) <br>
<code>-100.0</code>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; = lon of ref point (14.0, 9.0 (E is
positive in
GrADS, W is negative) <br>
<code>90000.0</code>&nbsp;&nbsp;&nbsp; = xref offset [m] <br>
<code>90000.0</code>&nbsp;&nbsp;&nbsp;  = yref offset [m]<br>
<code>14.0</code>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; = i of ref point
<br>
<code>9.0</code>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; = j of
ref
point <br>
<code>180000.0</code>&nbsp;  = dx [m] <br>
<code>180000.0</code>&nbsp;  = dy [m]</ul><p>

<i>Wind rotation has not been implemented!!!  Use only for scalar
fields.</i><p>

Source code in GrADS for the lon,lat -> i,j mapping:<p>
<pre>
<code>
void ll2ops(float *vals, float lni, float lti, float *grdi, float *grdj) 
   {

   const float radius = 6371229.0 ;   
   const float pi = 3.141592654;

   float stdlat, stdlon, xref, yref, xiref, yjref, delx , dely;

   float plt,pln;   
   double pi180,c1,c2,c3,c4,c5,c6,arg2a,bb,plt1,alpha,
   pln1,plt90,argu1,argu2;

   double hsign,glor,rstdlon,glolim,facpla,x,y;

   stdlat = *(vals+0);
   stdlon = *(vals+1);
   xref = *(vals+2);  
   yref = *(vals+3);
   xiref = *(vals+4);
   yjref = *(vals+5);  
   delx = *(vals+6);
   dely = *(vals+7);

   c1=1.0 ;
   pi180 = asin(c1)/90.0;

/* 
c 
c     set flag for n/s hemisphere and convert longitude to <0 ; 360> 
   interval 
c 
*/
   if(stdlat >= 0.0) {
     hsign= 1.0 ;  
   } else {
     hsign=-1.0 ;
   } 
/* 
c 
c     set flag for n/s hemisphere and convert longitude to <0 ; 360> 
   interval 
c 
*/  
   glor=lni ;
   if(glor <= 0.0) glor=360.0+glor ;
   rstdlon=stdlon;  
   if(rstdlon < 0.0) rstdlon=360.0+stdlon;

/* 
c 
c     test for a n/s pole case 
c 
*/
   if(stdlat == 90.0) {   
     plt=lti ;
     pln=fmod(glor+270.0,360.0) ;
     goto l2000;
   }

  if(stdlat == -90.0) {
     plt=-lti ;    
     pln=fmod(glor+270.0,360.0) ;
     goto l2000;
   }

/* 
c 
c     test for longitude on 'greenwich or date line' 
c 
*/  
   if(glor == rstdlon) {
     if(lti > stdlat) {      
       plt=90.0-lti+stdlat;
       pln=90.0;
     } else {
       plt=90.0-stdlat+lti;
       pln=270.0;;
     }   
     goto l2000;
   }

   if(fmod(glor+180.0,360.0) == rstdlon) {    
     plt=stdlat-90.0+lti;
     if(plt < -90.0) {
       plt=-180.0-plt;  
       pln=270.0;
     } else {
       pln= 90.0;
     }
     goto l2000;
   }

/* 
c 
c     determine longitude distance relative to rstdlon so it belongs to 
c     the absolute interval 0 - 180 
c 
*/
   argu1 = glor-rstdlon;
   if(argu1 > 180.0) argu1 = argu1-360.0;
   if(argu1 < -180.0) argu1 = argu1+360.0;

/* 
c 
c     1. get the help circle bb and angle alpha (legalize arguments) 
c 
*/

   c2=lti*pi180 ;
   c3=argu1*pi180 ;
   arg2a = cos(c2)*cos(c3) ;  
   if( -c1 > arg2a ) arg2a = -c1 ; /* arg2a = max1(arg2a,-c1)  */  
   if(  c1 < arg2a ) arg2a = c1 ; /* min1(arg2a, c1)         */
   bb = acos(arg2a) ;

   c4=hsign*lti*pi180 ;
   arg2a = sin(c4)/sin(bb) ;
   if( -c1 > arg2a ) arg2a = -c1 ; /* arg2a = dmax1(arg2a,-c1) */
   if(  c1 < arg2a ) arg2a = c1  ; /* arg2a = dmin1(arg2a, c1) */
   alpha = asin(arg2a) ; 
/* 
c 
c     2. get plt and pln (still legalizing arguments) 
c 
*/
   c5=stdlat*pi180 ;
   c6=hsign*stdlat*pi180 ;  
   arg2a = cos(c5)*cos(bb) + sin(c6)*sin(c4) ;
   if( -c1 > arg2a ) arg2a = -c1 ; /* arg2a = dmax1(arg2a,-c1) */
   if(  c1 < arg2a ) arg2a = c1  ; /* arg2a = dmin1(arg2a, c1) */
   plt1   = asin(arg2a) ;

   arg2a = sin(bb)*cos(alpha)/cos(plt1) ;

   if( -c1 > arg2a ) arg2a = -c1 ; /* arg2a = dmax1(arg2a,-c1) */
   if(  c1 < arg2a ) arg2a =  c1 ; /* arg2a = dmin1(arg2a, c1) */  
   pln1   = asin(arg2a) ;

/* 
c 
c    test for passage of the 90 degree longitude (duallity in pln) 
c         get plt for which pln=90 when lti is the latitude 
c 
*/   
   arg2a = sin(c4)/sin(c6);
   if( -c1 > arg2a ) arg2a = -c1 ; /* arg2a = dmax1(arg2a,-c1) */
   if(  c1 < arg2a ) arg2a =  c1 ; /* arg2a = dmin1(arg2a, c1) */
   plt90 = asin(arg2a) ;

/* 
c  
c         get help arc bb and angle alpha 
c 
*/
   arg2a = cos(c5)*sin(plt90) ;
   if( -c1 > arg2a ) arg2a = -c1 ; /* arg2a = dmax1(arg2a,-c1) */
   if(  c1 < arg2a ) arg2a =  c1 ; /* arg2a = dmin1(arg2a, c1) */
   bb    = acos(arg2a) ;

   arg2a = sin(c4)/sin(bb) ;
   if( -c1 > arg2a ) arg2a = -c1 ; /* arg2a = dmax1(arg2a,-c1) */
   if(  c1 < arg2a ) arg2a =  c1 ; /* arg2a = dmin1(arg2a, c1) */
   alpha = asin(arg2a) ;

/* 
c 
c         get glolim - it is nesc. to test for the existence of solution 
c 
*/
   argu2  = cos(c2)*cos(bb) / (1.-sin(c4)*sin(bb)*sin(alpha)) ;
   if( fabs(argu2) > c1 ) {    
     glolim = 999.0;
   } else {
     glolim = acos(argu2)/pi180;
   }

/* 
c 
c
     modify (if nesc.) the pln solution 
c 
*/
   if( ( fabs(argu1) > glolim && lti <= stdlat ) || ( lti > stdlat ) ) {   
     pln1 = pi180*180.0 - pln1;
   } 
/* 
c 
c     the solution is symmetric so the direction must be if'ed 
c 
*/
   if(argu1 < 0.0) { 
     pln1 = -pln1;
   } 
/* 
c 
c     convert the radians to degrees
c 
*/
   plt = plt1/pi180 ;
   pln = pln1/pi180 ;

/* 
c
c     to obtain a rotated value (ie so x-axis in pol.ste. points east) 
c     add 270 to longitude 
c 
*/  
   pln=fmod(pln+270.0,360.0) ;

 l2000:

/* 
c 
c     this program convert polar stereographic coordinates to x,y ditto 
c     longitude:   0 - 360  ; positive to the east 
c     latitude : -90 -  90  ; positive for northern hemisphere 
c     it is assumed that the x-axis point towards the east and 
c     corresponds to longitude = 0 
c 
c     tsp 20/06-89 
c 
c     constants and functions 
c 
*/
   facpla = radius*2.0/(1.0+sin(plt*pi180))*cos(plt*pi180);   
   x = facpla*cos(pln*pi180) ;   
   y = facpla*sin(pln*pi180)  ;

   *grdi=(x-xref)/delx + xiref;
   *grdj=(y-yref)/dely + yjref;

   return;

}
</pre></code></ul>
<br>
<br>

<a name="pit"><b><i>Pitfalls when using preprojected
data</i></b></a><p>
<ul>

There are a few <i>gotchas</i> with using preprojected data:<p>

<ol>
<li>the units in the variable definition for the <code>u</code> and <code>v</code>
components <b>must</b> be <code>33</code> and <code>34K</code> (the GRIB standard)
respectively, e.g.,<p>

<ul>
<code>u 15 33</code>&nbsp;&nbsp;&nbsp;  u component of the wind at 15 pressure levels
<br>
<code>v 15 34</code>&nbsp;&nbsp;&nbsp; v component of the wind at 15 pressure
levels</ul><p>

<li>wind rotation is handled for polar stereo (N and S)
preprojected data, but <i>not</i> for Lambert Conformal, as the Navy
rotates the winds relative to earth.  This will have to be added
later......

<li>the <code>eta.u</code> <b>and</b> <code>ops</code> projection are still
experimental...</ol>
</ul>
<br>
<br>

<a name="proj"><h2><u>GrADS Display Projections</u></h2></a>
<ul>
Now that you hopefully understand GrADS data grids, it is time to
discuss display projections. Graphics in GrADS are calculated
relative to the internal GrADS data grid <code>i,j</code> space, transformed
to the display device coordinates (e.g., the screen) and then
displayed.  That is, the i,j of the graphic element is converted
to <code>lat/lon</code> and then to <code>x,y</code> on the screen via a map
projection.<p>

GrADS currently supports four <code>display projections</code>:<p>

<ul>
<li>lat/lon (or spherical);
<li>N polar stereo (<a href="gradcomdsetmproj.html"><code>set mproj</a> nps</code>);
<li>S polar stereo (<a href="gradcomdsetmproj.html"><code>set mproj</a> sps</code>);
<li>the Robinson projection (set lon -180 180, set lat -90 90,
set mproj robinson).</ul><p>

As you can probably appreciate, the i,j-to-lon/lat-to-screen x,y
for <code>lon/lat</code> displays is very simple and is considerably more
complicated for N and S <code>polar stereo</code> projections.<p>

In principle, a Lambert Conformal display projection could be
implemented.  It just takes work and a simple user interface for
setting up that display projection.  Actually, the user interface
(i.e., "set" calls) is the most difficult problem...
</ul>
<br>
<br>

<a name="summary"><h2><u>Summary and Plans</u></h2></a>

<ul>
GrADS handles map projections in two different ways.  The first
is preprojected data where the fields are <i>already</i> on a projection
(e.g., Lambert Conformal).  It is fairly straightforward to
implement other preprojected data projections and we will be
fully implementing the NMC eta grid both staggered and
unstaggered, "thinned" gaussian grids and the CSU RAMS oblique
polar stereo projection.  The second is in how i,j graphics
(calculated in "grid" space) are displayed on a map background. 
Currently, only a few basic projections (lon/lat, polar stereo
and robinson) are supported, but perhaps the development group
will tackle this problem.</ul>


